Let K - group with normal subgroups N1 and N2. Intersection of which is trivial N1 \cap N2 = {e_k} and quotient groups of K by those normal groups are isomorphic to: K/N1 = N2, K/N2 = N1.
Is it true that in this case K = N1*N2? Where '*' gives us all posible products of N1 and N2 elements.
This is true for finite groups, but false in general. Truth for finite groups holds because of the isomorphism theorems.
For an infinite counterexample, consider the following: let W be a countably infinite set, W = X \cup Y, where X and Y are countably infinite and disjoint. Let X' and Y' be cofinite subsets of X and Y, respectively (I.e., the subset is obtained by removing a finite number of elements from the containing set). Let F be a field, and for each set Q let V(F, Q) denote a vector space with basis indexed by Q. Then V(F, Q) is, up to isomorphism, the same for Q = W, X, Y, X', or Y', yet we can set K:= V(F,W), N1:= V(F,X'), and N2:= V(F,Y'), but K N1*N2 even though the conditions you state are satisfied.
If I understood the Iaroslav Karkunov's question correctly, when he speaks about isomorphism K/N1 = N2 he means the isomorphism of canonical form that maps an element of N2 to its coset \in K/N1. In Geoffrey Mckenna's example, the isomorphism does not look this way.
Yes, Vladimir. You're right. That is the isomorphism I meant.
Geoffrey, сould you, please, show that in more detail in finite case?
I think you do not even need this much hypothesis for the conclusion. Just assume that N_1 is normal in K and $K/N_1 \cong N_2$ such that $N_1 \cap N_2 =1$. Then N_2 can be taken as a transversal (set of coset representatives). Thus $K \subseteq N_1N_2$.
In fact in Geoffrey Mckenna's example N1N2 is strictly smaller than K but isomorphic with K. (being a commutative example, it is in fact N1 + N2, a sum which is in this situation a direct sum). The question put by Iaroslav Karunov was about equality and not isomorphism, so Geoffrey's example is a good counterexample in the infinite case. This has been not so clear without Peter Breuer's proof and discussion.
Okay, but if I will reformulate the question as here:
https://www.researchgate.net/post/Is_there_a_natural_isomorphism_from_K_to_GxH_in_the_category_of_groups
Will it be true?
If I understood the Iaroslav Karkunov's question correctly, when he speaks about isomorphism K/N1 = N2 he means the isomorphism of canonical form that maps an element of N2 to its coset \in K/N1. In Geoffrey Mckenna's example, the isomorphism does not look this way.
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